Contents (60 sections)

EARLY UNIVERSE • FINITE-REGION QUANTUM DYNAMICS

Operational Separability Crossing

A constructive calculation of the first epoch at which two completed finite quantum records can coexist while conditioning on one produces less than one Granularity cell of influence in the other.

Pass-conditional • first-existence theorem

First operationally separable record pair

1.645530e-41 s

Controlling clockπħ/(2M_KK)

Massless envelope residual0.7335 ħ

Gapped KK residual0.1845 ħ

Nonlinear residual margin0.2665 ħ

Public claim. Under the explicitly stated finite-support and influence-envelope assumptions, the first existence of two operationally independent primitive 4D records occurs at 1.6455×10⁻⁴¹ s after the initial boundary. The root is fixed by record completion; geometric separation is already possible earlier.

What this does not say

The theorem is existential, not universal. Adjacent records, global/topological sectors and long-range correlations can remain interdependent after this time. Exact universal factorization is neither assumed nor claimed.

THEOREM

Result in one line

The finite record support has localization clock τ_KK=ħ/M_KK, while creation of a perfectly distinguishable completed record is bounded by the Margolus–Levitin time πħ/(2M_KK). Because π/2>1, record completion is the active constraint:

Operational-separability root

t_sep^∃ = πħ/(2M_KK) = 1.645530e-41 s

At that root, the conservative massless influence envelope is 0.733471ħ and the gapped KK envelope is 0.184518ħ. Both lie below the one-action-cell Granularity boundary and decrease under the modeled local kernels.

Why the claim is assertive

The numerical root is not selected by an arbitrary ε and is not tuned to standard cosmology. It follows from the frozen compactification scale, the finite-record energy ceiling and a quantum speed limit. The remaining conditionality is isolated to the stated influence/tower/global-sector bounds.

SCOPE

Claim boundary: existence versus universal factorization

There are two different questions and they must not be conflated.

$t_{\rm sep}^{\exists}=\inf\{t:\exists\;A,B\ {\rm operationally\ independent}\}$

$t_{\rm sep}^{\forall}=\inf\{t:\forall\;A,B\ {\rm operationally\ independent}\}$

This dossier computes a candidate for t_sep^∃: the first time at least one pair of separately retrievable primitive records can coexist while conditioning on one is sub-Granular in the other. It does not claim t_sep^∀; global constraints, topological sectors, shared gauge data, adjacent supports, and long-lived entanglement can continue later.

Scientific interpretation

The result is a proposed onset of a recordable, subsystem-resolved universe, not the disappearance of entanglement and not a literal collapse of a universal wavefunction into a product state.

AUTHORITY

Source authority and branch discipline

Authority	File	SHA-256
Interdependence 4.0	BB_INT_4_0_MAX_RIGOR_ALL_GATE_INTERDEPENDENCE.md	54cfaba68331084195…
Granularity 4.0	BB_GRN_4_0_MAX_RIGOR_ALL_GATE_GRANULARITY.md	9546a8071ec4855e5e…
Dynamics 4.2	BB_DYN_4_2_MAX_RIGOR_FULL_GATE_CLOSURE_DYNAMICS.md	cbc41bd5faf197208d…
Observer 4.0	BB_OBS_4_0_MAX_RIGOR_ALL_GATE_OBSERVER.md	2a7edf478134d753cf…
Scale 4.1	BB_SCL_4_1_MAX_RIGOR_ALL_GATE_SCALE.md	b0426790624d7abde9…
Shape Stage	SHAPE_STAGE.md	20a1906b5467a1d82b…
Shape Actor	SHAPE_ACTOR.md	88b684f610f46ea28b…
Shape Rulebook	SHAPE_RULEBOOK.md	ff0e5a660e11fbd1a3…
GUT authority	GUT.md	f1fb418c93f004afac…
Granularity root dossier	05_ROOT_GRANULARITY_dossier.md	b242e7e207f78156d5…

Interdependence owns factorization and Schur complements; Dynamics owns causal/open quantum evolution and finite history; Observer owns instruments and accessible algebras; Granularity owns the operational quotient. Any changed authority or ruler creates a new branch.

ANCHORS

What is anchored and what is not

Object	Role	Status
ħ	action quantum / conversion	measured anchor
R₆	compact Stage radius	frozen Shape
M_KK=1/R₆	13D→4D correlation/threshold scale	derived from Shape
M_*	13D normalization scale	secondary cross-check only
H₀	cosmological validation	forbidden derivation input
13.8 Gyr age	external validation only	forbidden derivation input

The central crossing uses M_KK, not M*. This is important: M_KK is the scale actually associated with compactification and heavy-mode separation, and it is unaffected by the unresolved factor-of-two R_Y interval-volume convention.

ONTOLOGY

No point mass — finite causal supports are primitive

$A,B\subset\mathcal P_{\rm phys},\qquad \mathcal A_A,\mathcal A_B\subset\mathcal A_{\rm phys}$

A and B are finite causal record supports or Actor-owned subalgebras. They may be wave packets, gauge-invariant local field algebras, finite mode bands, boundary sectors, or other lawful supports. No delta-function source and no point-particle ontology enters the derivation.

This also avoids importing a classical particle picture into an epoch where a particle interpretation may not yet be valid.

INTERDEPENDENCE

Factorization firewall

The current Interdependence authority explicitly forbids promoting product Stage geometry into an unproved tensor-product Hilbert space.

$\mathcal H_{\rm phys}\neq\mathcal H_A\otimes\mathcal H_B\quad{\rm unless\ separately\ certified}$

The primary formulation therefore uses restrictions of states to accessible observable algebras. Trace-distance formulas appear only as a specialization when a factorization packet has been discharged.

OBSERVER

Completed record operation in A

$\mathcal M_A=\{\mathcal M_a\},\qquad\sum_a\mathcal M_a\ {\rm CPTP}$

$\omega_{B|a}(O_B)=\frac{\omega\!\left(\mathcal M_a^\dagger[O_B]\right)}{p(a)}$

“Collapse” is used only as shorthand for conditioning on a completed, normalized physical record. Dynamics DYN-C13 and Observer OBS-C12 require the instrument, environment, outcome algebra, normalization and composition law to be explicit.

DEFINITION

Conditional influence, not no-signaling

$\mathcal I_{A\to B}(t)=\sup_{\mathcal M_A,a}D_{\mathcal A_B}(\omega_{B|a,t},\omega_{B,t})$

If the A outcome is discarded, B's marginal can remain unchanged even for a strongly entangled state. That no-signaling identity is not the clock. The clock is the conditioned record difference.

S_IF DERIVATION

The parent quadratic problem

Expand the complete physical parent action around the frozen background and quotient, retaining one B-sector and an environmental/heavy sector H. The mixed block is not allowed to disappear by assumption.

$S^{(2)}=\frac12\,B Q_{BB}B+B V H+\frac12\,H Q_{HH}H+J_A C H$

J_A represents the branch/record operation sourced from A. Q_HH contains the KK/internal spectrum, while V and C carry the lawful mixed couplings.

SCHUR COMPLEMENT

Exact Gaussian integration and Schur complement

At quadratic order, H can be integrated exactly. This is the controlled realization of INT-C06/INT-C11.

$Q_{B,{\rm eff}}=Q_{BB}-VQ_{HH}^{-1}V^\dagger$

$S_{AB,{\rm eff}}=-B\,VQ_{HH}^{-1}C^\dagger J_A$

The second term is the explicit branch-conditioned A→B influence channel. Every omitted nonlinear term is kept in the residual ledger rather than folded into this Gaussian result.

INFLUENCE FUNCTIONAL

Closed-time-path influence functional

For an open B-sector the correct history object is the closed-time-path influence action, which preserves memory and noise rather than forcing a Markov approximation.

$e^{\frac{i}{\hbar}S_{\rm IF}[B_+,B_-;J_A]}=\int\mathcal D H_+\mathcal D H_-\;e^{\frac{i}{\hbar}(S_H[H_+]-S_H[H_-]+S_{\rm mix})}\,\rho_H$

$S_{\rm IF}=-\!\int B_-D_RB_++\frac{i}{2}\!\int B_-N B_-+S_{AB}[J_A,B_\pm]$

D_R is the retarded reaction kernel and N the noise/Hadamard kernel. This is the requested S_IF at the first controlled (quadratic/Gaussian) level.

HISTORY CHANNEL

History channel Φ_history

$\rho_B(t)=\Phi_{t:0}^{[J_A]}[\rho_B(0)]$

$\rho_B(B_f^+,B_f^-;t)=\!\int\!\mathcal D B_+\mathcal D B_-\;e^{\frac{i}{\hbar}(S_B^+-S_B^-+S_{\rm IF})}\rho_B(0)$

This path-integral representation is the non-Markovian history channel. Composition is a process tensor when environmental memory crosses intermediate time cuts. A Lindblad semigroup is therefore not assumed.

COMMUTATION

Parent-to-record commutation

Interdependence INT-C09 requires the parent reduction and the Observer record map to commute within declared error bounds.

$\Pi_{\rm obs}\circ\Phi_{\rm parent}\simeq\Phi_{\rm record}\circ\Pi_{\rm obs}$

The separability result is invalid if a basis change, projection, gauge-owner reassignment or calibration change alters the record without restarting the branch.

q_B

Constructing q_B from the Granularity cost floor

Granularity 4.0 forbids using a non-transitive pairwise epsilon rule as the quotient. The older root dossier supplies the gate-relevant cost currency: stable independently retrievable records require a positive action/cost cell Δ₀, whose measured residue is ħ.

$\mathcal C_{A\to B}(t)=\sup_{\mathcal M_A,a,h_B}\left|\Delta S_{\rm IF}^{(a)}[h_B;t]\right|$

For this gate, zero conditional influence is a canonical origin. Freeze the deterministic action-cell quantizer

$q_B(\mathcal C)=\left\lfloor\frac{\mathcal C}{\hbar}\right\rfloor,\qquad \mathcal C\ge0$

with exact-invariant and Observer-signature components appended as required by GRN-4.0. In particular, conditioning on A is sub-Granular in B whenever 0≤C<ħ, because both the conditioned and unconditioned influence costs lie in the zero-action cell.

Status of q_B

This is a calculation-local frozen quantizer built from the framework’s declared action currency and ħ residue. It is not claimed to be the unique possible quantizer for every experiment. Wrong-offset and refinement controls are included later.

GRANULARITY

Why the old arbitrary epsilon disappears

The feasibility estimate used ε=10⁻³, 10⁻⁶ and similar values only to test scale stability. The full gate instead compares an influence action with the declared action cell.

$\mathcal C_{A\to B}<\hbar\quad\Longrightarrow\quad q_B(\omega_{B|a})=q_B(\omega_B)$

The remaining uncertainty is therefore physical—whether the influence kernel obeys the stated finite-region envelope—not an arbitrary numerical tolerance.

SCALE

Why M_KK is the correct central clock

$R_6=1.591549430918954\times10^{-17}\ {\rm GeV}^{-1}$

$M_{\rm KK}=\frac1{R_6}=6.283185307179586\times10^{16}\ {\rm GeV}$

The physical question concerns the loss of operational influence associated with the common higher-dimensional state. M_KK is precisely the scale at which internal excitations cease to behave as unresolved 4D structure. It is therefore more directly relevant than M*, which normalizes the full 13D gravitational action.

Using M_KK also removes the current R_Y factor-of-two volume ambiguity from the central number.

PROTOCOL

The record protocol scale is not a minimum length

$\tau_{\rm KK}=\frac{\hbar}{M_{\rm KK}}={tau_KK:.12e}\ {\rm s}$

$\ell_{\rm KK}=c\tau_{\rm KK}=\frac{\hbar c}{M_{\rm KK}}={ell_KK:.12e}\ {\rm m}$

ℓ_KK is the support diameter chosen for this UV/compactification-matched record protocol. Granularity explicitly does not assert a universal smallest spacetime length. The protocol simply uses the same energy and causal time scale on one ruler.

RECORD COMPLETION

Minimum completion time for one distinguishable record

As a downstream quantum confirmation of the Granularity action floor, Margolus–Levitin gives the fastest orthogonalization of a state with energy E above its ground state.

$\tau_{\rm ML}\ge\frac{\pi\hbar}{2E}$

For the earliest 4D/KK-resolved record we take the limiting energy E→M_KK⁻. Then

$t_{\rm rec,min}=\frac{\pi\hbar}{2M_{\rm KK}}={t_rec:.12e}\ {\rm s}$

Why this is a lower bound

Any record built at a lower characteristic energy takes longer. Therefore this is the earliest completion time permitted by the chosen compactification-scale record protocol, not a tuned epoch.

NORMALIZATION

Dimensionless clock

$x\equiv\frac{M_{\rm KK}t}{\hbar}$

The first complete record occurs at

$x_{\rm rec}=\frac\pi2=1.57079632679$

This dimensionless representation makes the crossing transparent and separates the geometry from unit conversion.

CAUSAL GEOMETRY

Causal packing of two finite record supports

At proper time t, a causal domain has diameter 2ct. Each primitive record support has protocol diameter ℓ_KK. Two non-overlapping cells can first fit when x≥1.

$2ct\ge2\ell_{\rm KK}\quad\Longleftrightarrow\quad x\ge1$

If both cells remain fully inside the causal domain, their maximum center separation is

$r_{\max}(t)=2ct-\ell_{\rm KK}=(2x-1)\ell_{\rm KK}$

At the record-completion time, s_max=r_max/ℓ_KK=π−1=2.141592654.

MASSLESS ENVELOPE

Massless sector: deliberately slow 1/r envelope

To avoid overclaiming, take a 1/r falloff for the branch-conditioned record amplitude. This is slower than the action/flux falloff of ordinary radiative records and is therefore conservative for a finite-energy local signal.

$F_0(r)\le\frac{\ell_{\rm KK}}r$

The maximum action available to one primitive transition over time t is bounded by M_KKt. At the farthest allowed separation,

$\frac{\mathcal C_0(t)}\hbar\le x\,\frac1{2x-1}$

The geometric cell crossing solves x/(2x−1)=1, hence

$x_{\rm geom}=1$

For every x>1 the massless influence envelope is already below one action cell.

MASSLESS RESULT

Massless residual at first record completion

$\left.\frac{\mathcal C_0}{\hbar}\right|_{x=\pi/2}=\frac{\pi/2}{\pi-1}={massless_ratio:.12f}$

The margin to the one-cell threshold is 26.653%.

Meaning

Even under the deliberately slow 1/r amplitude envelope, the far record cannot receive enough branch-conditioned action to occupy a different Granularity action cell by the time the first record can complete.

GAPPED ENVELOPE

Heavy/KK sector envelope

For a gapped equal-time/spacelike correlation channel, use the conservative normalized clustering envelope

$F_H(r)\le e^{-M_{\rm KK}r/(\hbar c)}$

With r=r_max,

$\frac{\mathcal C_H(t)}\hbar\le x\,e^{-(2x-1)}$

For x≥1 this function is monotonically decreasing because d ln f/dx = 1/x−2<0. Therefore it is already sub-Granular when two cells first fit.

HEAVY RESULT

Heavy residual at first record completion

$\left.\frac{\mathcal C_H}{\hbar}\right|_{x=\pi/2}=\frac\pi2\,e^{-(\pi-1)}={heavy_ratio:.12f}$

The gapped contribution is less than one quarter of a record quantum in this conservative envelope.

COMPOSITION

Mixed massless + heavy channel bound

Let w₀ and w_H be the fractions of the primitive influence-action budget carried by massless and gapped sectors, with nonnegative weights summing to one.

$\frac{\mathcal C_{\rm tot}}\hbar\le w_0\frac{x}{2x-1}+w_H x e^{-(2x-1)},\qquad w_0+w_H=1$

At x=π/2, every convex mixture is bounded by the larger component:

$\frac{\mathcal C_{\rm tot}}\hbar\le {massless_ratio:.12f}<1$

Thus no tuning of the massless/heavy partition is needed for the first-existence result.

ROOT FIND

Root finding: which condition actually controls?

There are two necessary inequalities:

$x\ge x_{\rm geom}=1\qquad{\rm and}\qquad x\ge x_{\rm rec}=\frac\pi2$

The operational-separability existence crossing is their maximum:

$x_{\rm sep}^{\exists}=\max\!\left(1,\frac\pi2\right)=\frac\pi2$

$t_{\rm sep}^{\exists}=\frac{\pi\hbar}{2M_{\rm KK}}={t_rec:.12e}\ {\rm s}$

Central result

Within the frozen finite-region protocol and influence envelopes, the conditional-influence condition becomes satisfiable before a stable distinguishable record can complete. Record completion is therefore the load-bearing clock.

PERSISTENCE

Persistence after the crossing

For x≥π/2, the massless envelope x/(2x−1) decreases monotonically toward 1/2. The gapped envelope x exp[−(2x−1)] also decreases monotonically for x≥1.

$\frac{d}{dx}\frac{x}{2x-1}=-\frac1{(2x-1)^2}<0$

$\frac{d}{dx}\left[x e^{-(2x-1)}\right]=e^{-(2x-1)}(1-2x)<0\quad(x>1/2)$

Therefore the crossing is persistent under the stated envelope: once the first record completes, these local influence bounds do not revive above the Granularity cell threshold.

NUMERICS

Numerical geometry at the crossing

Quantity	Value
t_sep^∃	1.645529892250e-41 s
τ_KK	1.047576865428e-41 s
ℓ_KK	3.140556433606e-33 m
causal radius c t	4.933174511101e-33 m
causal diameter	9.866349022202e-33 m
max center separation r_max	6.725792586394e-33 m
r_max/ℓ_KK	2.141592653590

All lengths here are protocol/support scales, not claims of fundamental spacetime discreteness.

R_Y CONFLICT

Why M* is now only a secondary cross-check

The previous dossier carried two M* branches because the R_Y interval convention changes the 9D volume. If one nevertheless asks for an M*-limited orthogonalization time, the branches are

$t_{M_*,A}=1.30008025948\times10^{-41}\ {\rm s}$

$t_{M_*,B}=1.38463889592\times10^{-41}\ {\rm s}$

These bracket below the M_KK-controlled separability time of 1.646e-41 s. The close clustering is a useful cross-check, but M* is not used to set the central result.

SENSITIVITY

Support-size sensitivity

Let the primitive record-support diameter be λℓ_KK instead of exactly ℓ_KK. Causal packing gives s_max=2x−λ, and the conservative massless crossing becomes

$x_{\rm geom}(\lambda)=\frac{\lambda}{2-\lambda},\qquad0<\lambda<2$

The record-completion root π/2 remains controlling whenever

$\lambda\le\frac{2\pi}{\pi+2}=1.2220309407\ldots$

Thus the central result is stable to roughly +22% enlargement of the canonical support diameter. Larger protocol cells move the crossing later and are reported in the sensitivity table rather than silently absorbed.

SENSITIVITY

Support-size sensitivity table

λ	x_geom	x_sep	t_sep (s)
0.5000	0.333333	1.570796	1.645530e-41
0.7500	0.600000	1.570796	1.645530e-41
1.0000	1.000000	1.570796	1.645530e-41
1.1000	1.222222	1.570796	1.645530e-41
1.2000	1.500000	1.570796	1.645530e-41
1.2220	1.570796	1.570796	1.645530e-41
1.3000	1.857143	1.857143	1.945500e-41
1.5000	3.000000	3.000000	3.142731e-41
1.7500	7.000000	7.000000	7.333038e-41
1.9000	19.000000	19.000000	1.990396e-40

The table is a protocol deformation audit, not a parameter fit. λ=1 is the frozen central protocol because it identifies one support diameter with one light-crossing length at the KK action/time scale.

ROBUSTNESS

Tail-exponent sensitivity

Replace the deliberately slow massless envelope by F∝(ℓ/r)^p. For p≥1,

$\frac{\mathcal C_p}{\hbar}\le x(2x-1)^{-p}$

p=1 is the slowest case carried in the central calculation. Any p>1—such as energy-flux or gauge-invariant local-correlation falloff—crosses earlier and increases the safety margin.

p	C/ħ at x=π/2
1	0.733471
1.5	0.501204
2	0.342489
3	0.159922
4	0.074675

ROBUSTNESS

Energy-fraction sensitivity

If the primitive record transition uses E=ηM_KK with 0<η≤1, then the minimum completion time is

$t_{\rm rec}(\eta)=\frac{\pi\hbar}{2\eta M_{\rm KK}}$

Lower energy makes the first record later, enlarges the available causal diameter, and cannot make the earliest crossing earlier than the η=1 result. Thus η=1 gives the legitimate lower envelope for the oldest possible record.

NEGATIVE CONTROL

Wrong-quantizer control

Granularity requires a frozen deterministic cell map. A malicious offset

$q_B^{(\delta)}(C)=\left\lfloor\frac{C+\delta}{\hbar}\right\rfloor$

would move the zero-cell boundary if δ were selected after seeing the answer. This control is rejected because the no-influence state C=0 is the physical origin and the framework’s cost floor is measured from zero action. The central q_B therefore fixes δ=0 before root finding.

ABLATION

Interaction-off ablation

Set the mixed blocks V=C=0. Then the influence functional loses its A–B cross term:

$S_{AB,{\rm eff}}=0\quad\Longrightarrow\quad\mathcal C_{A\to B}=0$

Operational independence is immediate once both records exist. Restoring the mixed blocks restores the finite residuals. This is the required causal ablation/restoration signature.

NEGATIVE CONTROL

No-gap destructive control

Set the heavy gap artificially to zero while keeping a constant long-range influence envelope. Then the heavy suppression disappears and a shared/global mode can remain above one action cell.

$M_{\rm gap}\to0\quad\not\Rightarrow\quad\mathcal C_H<\hbar$

The calculation must therefore retain genuinely global/topological/shared-resource modes as separate rows. They are not allowed to inherit the local-sector crossing automatically.

CLAIM CEILING

Adjacent-cell control: why this is an existence crossing

At x=π/2, two adjacent cells separated by one support diameter do not satisfy the conservative 1/r record-amplitude bound:

$\frac{\mathcal C_{\rm adjacent}}\hbar\lesssim x_{\rm rec}=\frac\pi2>1$

So the result does not mean every neighboring subsystem is already independent. It means the causal domain is large enough to contain some pair whose conditional coupling is sub-Granular. That distinction is essential.

GLOBAL MODES

Global-sector firewall

Some constraints and topological/global variables are not expected to attenuate like local radiation. Interdependence INT-C08, INT-C17 and INT-C21 require them to remain explicitly owned.

If a lawful global record changes B after conditioning on A, that mode belongs to the universal/all-subsystems problem t_sep^∀, not to the local first-existence certificate. The local result survives only because it is typed narrowly and does not erase those global rows.

NONLINEAR RESIDUAL

Higher-order interaction residual

The displayed S_IF is exact for the Gaussian/quadratic parent sector. Cubic, quartic, Yukawa and nonperturbative interactions generate higher cumulants:

$S_{\rm IF}=S_{\rm IF}^{(2)}+S_{\rm IF}^{(3)}+S_{\rm IF}^{(4)}+\cdots$

The current gate can close only conditionally unless those terms are either bounded below the one-cell margin or included explicitly. The central massless margin is about 0.267 ħ, which provides a quantitative target for the nonlinear remainder.

$\|S_{\rm IF}^{(\ge3)}\|<({1-massless_ratio:.12f})\hbar\quad\Rightarrow\quad{\rm crossing\ preserved}$

TOWER SCOPE

Finite-tower versus infinite-tower residual

The framework’s causality/locality work is strongest at finite KK truncation and explicitly keeps the full infinite-tower nonperturbative completion open. Therefore this dossier states:

Finite admitted tower: the gap/locality envelope can be tested mode by mode.
Infinite tower: a uniform summability/cluster bound is still owed.
Local first-existence crossing: conditional until that tail is bounded or a finite physical cutoff is shown to be the correct record scope.

MEMORY

History-channel residual and memory

The process-tensor form of Φ_history permits non-Markovian memory. Persistence requires that later memory kernels not restore an above-cell difference.

$\sup_{t\prime\ge t_{\rm sep}}\mathcal C_{A\to B}(t\prime)<\hbar$

The monotone envelope proves this for the modeled local kernels, but a full nonperturbative history-channel certificate must include every retained memory term.

VALIDATION

External validation firewall

Nothing on this page is used upstream in the derivation. It is comparison only.

Benchmark	Externally established statement	Use here
Standard hot-universe mapping	T=M_KK maps to t≈5.930e-41 s for g*=106.75	order-of-magnitude check only
Equivalent standard temperature	t=1.646e-41 s maps to T≈1.193e+17 GeV	order-of-magnitude check only
Electroweak crossover	T_c≈159.5±1.5 GeV	downstream future validation
QCD crossover	T_c≈156.5±1.5 MeV	downstream future validation
BBN	T≈1 MeV corresponds to t≈1 s	strong observed-history check
Planck base ΛCDM	age≈13.8 Gyr	late-time comparison only

The Shape crossing and conventional high-temperature extrapolation differ by a factor of about 3.60, while occupying the same 10⁻⁴¹-second decade.

INTERPRETATION

Why “same decade” is the appropriate current comparison

No direct observation probes 10⁻⁴¹ s. The standard value at such an epoch is itself an extrapolation of GR + thermal field theory. Therefore a factor-of-few comparison is a consistency check, not a precision test.

The meaningful future test is whether the Shape-derived history, propagated forward without importing the standard expansion curve, reaches the empirically secure BBN and CMB record constraints.

FORWARD PROGRAM

Connection to electroweak, QCD, BBN and CMB

The next history calculation should propagate the post-separability state through a sequence of finite record transitions:

$t_{\rm sep}\to T_{\rm EW}\to T_{\rm QCD}\to\nu\ {\rm decoupling}\to{\rm BBN}\to{\rm recombination}$

Each transition must consume the reconstructed matter content, actual Dynamics and Observer record map. The externally known transition temperatures remain blind validation targets.

COSMIC AGE

Implication for the “age of the recordable universe”

Under the present finite-region protocol, the age of the recordable universe starts not at an exact singular coordinate but at the first epoch at which (i) stable records can complete and (ii) at least two such records can be operationally independent.

$t_{\rm recordable,start}^{\exists}\approx1.6455\times10^{-41}\ {\rm s}$

The present-day age of the recordable universe would then be t_today−t_{recordable,start}; because the start is microscopic, determining the 13.8-Gyr-scale age is overwhelmingly a late-time history/expansion problem, not sensitive to subtracting 10⁻⁴¹ s.

CLAIM CEILING

What this calculation does not yet give

It does not derive the present cosmic age.
It does not prove universal factorization of the wavefunction.
It does not prove all adjacent regions are independent at t_sep.
It does not close the infinite-KK-tower nonperturbative clustering problem.
It does not derive ħ.
It does not use a point mass.

Honest terminal

The strongest current status is PASS-CONDITIONAL: first-existence operational-separability crossing, with nonlinear/infinite-tower/history-tail bounds still owed for an unconditional certificate.

CLOSURE PATH

Minimal closing conditions for unconditional status

Compute or bound S_IF^(≥3) so the nonlinear remainder stays below the 0.267 ħ safety margin.
Prove a uniform finite-record tail bound for the admitted KK tower, or freeze a finite physical record cutoff.
Enumerate every massless/global Actor-owned channel and show each either obeys the 1/r-or-faster finite-record envelope or is typed outside the local-existence claim.
Regenerate the Observer/Granularity packets with the q_B hash and wrong-quantizer controls.
Run an independent review agent on the branch and destructive controls.

RESULT LEDGER

Machine-checkable result ledger

Item	Value / status
M_KK	6.2831853071795840e+16 GeV
τ_KK	1.0475768654282460e-41 s
ℓ_KK	3.1405564336057557e-33 m
x_geom	1.0
x_record	π/2
t_sep^∃ candidate	1.6455298922500005e-41 s
massless residual	0.7334711034621300 ħ
heavy residual	0.1845176883464771 ħ
q_B	floor(C/ħ), zero-origin
S_IF	Gaussian exact; nonlinear remainder OPEN
Φ_history	CTP/process-tensor construction
universal t_sep^∀	OPEN

CONTROLS

Falsification suite

Test	Expected result
V=C=0 ablation	C_A→B=0
restore mixed blocks	finite influence returns
remove KK gap	heavy bound must weaken/fail
insert constant global mode	local-envelope proof must refuse promotion
offset q_B after seeing target	FAIL calibration-lineage control
λ support enlargement	crossing moves according to λ/(2−λ)
include nonlinear remainder above margin	crossing reopens
later memory revival above ħ	persistence fails

REPRODUCIBILITY

Reproducibility arithmetic

M_KK = 1 / R6
      = 1 / (1.591549430918954e-17 GeV^-1)
      = 6.2831853071795840e+16 GeV

tau_KK = hbar / M_KK
       = 1.0475768654282460e-41 s

t_sep = (pi/2) tau_KK
      = 1.6455298922500005e-41 s

s_max(x_rec) = 2*(pi/2)-1 = pi-1 = 2.1415926535897931
massless = (pi/2)/(pi-1) = 0.7334711034621300
heavy = (pi/2)*exp(-(pi-1)) = 0.1845176883464771

The accompanying JSON and CSV files contain the same values in machine-readable form.

REFERENCES

External references used only after freeze

Particle Data Group, 2024/2025 update, Big-Bang Nucleosynthesis: BBN is the deepest reliable Standard-Model probe of the early universe; T~1 MeV, t~1 s.
D’Onofrio & Rummukainen, The Standard Model cross-over on the lattice, arXiv:1508.07161: T_c=159.5±1.5 GeV.
Steinbrecher et al., The QCD crossover..., arXiv:1807.05607: T_c=156.5±1.5 MeV.
Planck Collaboration 2018 VI, arXiv:1807.06209: base-ΛCDM late-time cosmological parameters and ~13.8 Gyr inferred age.

These references are validation data, not inputs to the crossing calculation.

SOURCE MAP

Project-source excerpts: what each block contributes

Block	Load-bearing contribution
Granularity 4.0	deterministic record-cell quotient; dynamic/history/region/test-family completeness
Granularity root	positive action/cost floor Δ₀ with measured ħ residue; no universal minimum length
Interdependence 4.0	mixed block matrix, Schur complement, factorization taxonomy, decoupling bounds, quantum composition
Dynamics 4.2	CPTP/open quantum evolution, nonequilibrium history, causal response
Observer 4.0	instruments, accessible algebra, temporal/regional record supports, operational distinguishability
Shape Stage	R₆ and compactification geometry; explicit R_Y convention conflict
GUT authority	M_KK/KK-spectrum context and M* cross-check

SYNTHESIS

Final derivation in one chain

$R_6\to M_{\rm KK}=R_6^{-1}\to \tau_{\rm KK}=\hbar/M_{\rm KK}\to t_{\rm record}=\pi\tau_{\rm KK}/2$

$2ct\ {\rm causal\ diameter}\to r_{\max}=(2x-1)\ell_{\rm KK}\to \mathcal C_{A\to B}<\hbar$

$\max(x_{\rm geom},x_{\rm record})=\max(1,\pi/2)=\pi/2$

Candidate first-existence crossing

1.645530e-41 s

At the crossing, the worst carried local massless envelope is 0.733 action quanta and the gapped envelope is 0.185. The conditional influence remains below the frozen zero-cell boundary thereafter under the modeled local kernels.

VERDICT

Derivation verdict

PASS-CONDITIONAL — FIRST-EXISTENCE OPERATIONAL SEPARABILITY

The source-grounded finite-region calculation produces a non-arbitrary candidate onset time

t_sep^∃ = 1.645530e-41 s

controlled by the Margolus–Levitin completion time at the Shape KK scale. The local conditional-influence envelope is already sub-Granular by then.

Still OPEN for unconditional closure

Nonlinear influence cumulants, the infinite-tower uniform tail, every genuine global/shared mode, and the full all-subsystem t_sep^∀ certificate.

Bottom line

The full calculation strengthens the earlier order-of-magnitude result: the 10⁻⁴¹-second epoch is not merely where several scales happen to cluster. Under a finite-record causal protocol, it is the first point at which record completion and operational separability can simultaneously be satisfied.

HOSTILE REVIEW

Reviewer attack 1 — does q_B smuggle the answer?

No target epoch enters q_B. The quantizer is fixed before root finding by the framework’s action-cost currency, the zero-influence origin, and the measured ħ residue. The crossing moves under declared protocol deformations and those movements are published.

HOSTILE REVIEW

Reviewer attack 2 — is 1/r chosen to make it pass?

No. 1/r is intentionally the slowest local far-field amplitude envelope carried in the central calculation. Ordinary record action/energy deposition falls at least as fast as amplitude squared, and gapped correlations fall faster. If an admitted local channel is shown to fall more slowly than 1/r, this gate must reopen.

HOSTILE REVIEW

Reviewer attack 3 — what about gravity?

A finite gravitational-wave strain can scale as 1/r, which is why the central envelope uses 1/r rather than 1/r². The calculation does not require a point mass: A may be any finite stress-energy wave packet. A nondecaying global gravitational mode would be typed as a global/shared sector and excluded from automatic local promotion.

HOSTILE REVIEW

Reviewer attack 4 — what about entanglement?

Entanglement may persist. The claim is only that conditioning on A does not have enough operational action budget to move the selected B record into a different frozen Granularity cell. Mathematical nonfactorization and operational independence are not the same statement.

HOSTILE REVIEW

Reviewer attack 5 — what if M_KK changes?

The time scales inversely with the frozen KK scale:

$\frac{\delta t_{\rm sep}}{t_{\rm sep}}=-\frac{\delta M_{\rm KK}}{M_{\rm KK}}=\frac{\delta R_6}{R_6}$

Any future Shape update to R₆ therefore invalidates this certificate and regenerates the crossing exactly as Interdependence requires.